Model reduction for the material point method via an implicit neural representation of the deformation map

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates kinematics by approximating deformation map using an implicit neural representation that restricts trajectories to reside low-dimensional manifold. By explicitly map, its spatiotemporal gradients—in particular gradient and velocity—can be computed via analytical differentiation. In contrast typical techniques construct linear or manifold approximate (finite number of) degrees of freedom characterizing given spatial discretization, use enables proposed continuous map. allows kinematic approximation remain agnostic discretization. Consequently, supports dynamic discretizations—including resolution changes—during course online reduced-order-model simulation. To generate dynamics generalized coordinates, we propose family projection techniques. At each time step, these techniques: (1) Calculate full-space at quadrature points, (2) subset ‘sample’ (3) reduced-space projecting updated position velocity onto tangent space, respectively. We achieve significant computational speedup hyper-reduction ensures all three steps execute only small problem's domain. Large-scale numerical examples with millions points illustrate method's ability gain order magnitude computational-cost saving—indeed real-time simulations—with negligible errors.